WORST_CASE(Omega(n^1),O(n^3))
proof of input_CL0HxVq59e.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 297 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 696 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 216 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 2104 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^3)
(35) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
    (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxRelTRS
    (39) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) typed CpxTrs
    (43) OrderProof [LOWER BOUND(ID), 9 ms]
    (44) typed CpxTrs
    (45) RewriteLemmaProof [LOWER BOUND(ID), 412 ms]
    (46) BEST
        (47) proven lower bound
            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
            (49) BOUNDS(n^1, INF)
        (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 49 ms]
            (52) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
gcd :: 0:s -> 0:s -> 0:s
if_gcd :: true:false -> 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   gcd_2
   if_gcd_3

(c) The following functions are completely defined:

   le_2
   minus_2
   if_minus_3

Due to the following rules being added:

   if_minus(v0, v1, v2) -> 0 [0]

And the following fresh constants: none

----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
gcd :: 0:s -> 0:s -> 0:s
if_gcd :: true:false -> 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), 0) -> if_minus(false, s(x), 0) [2]
   minus(s(x), s(y')) -> if_minus(le(x, y'), s(x), s(y')) [2]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(0)) -> if_gcd(true, s(x), s(0)) [2]
   gcd(s(0), s(s(x'))) -> if_gcd(false, s(0), s(s(x'))) [2]
   gcd(s(s(y'')), s(s(x''))) -> if_gcd(le(x'', y''), s(s(y'')), s(s(x''))) [2]
   if_gcd(true, s(0), s(y)) -> gcd(0, s(y)) [2]
   if_gcd(true, s(s(x1)), s(y)) -> gcd(if_minus(le(s(x1), y), s(x1), y), s(y)) [2]
   if_gcd(false, s(x), s(0)) -> gcd(0, s(x)) [2]
   if_gcd(false, s(x), s(s(x2))) -> gcd(if_minus(le(s(x2), x), s(x2), x), s(x)) [2]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
gcd :: 0:s -> 0:s -> 0:s
if_gcd :: true:false -> 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 1
   false => 0

----------------------------------------

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   gcd(z, z') -{ 2 }-> if_gcd(le(x'', y''), 1 + (1 + y''), 1 + (1 + x'')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y''), y'' >= 0, x'' >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + x')) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
   gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + x1, y), 1 + x1, y), 1 + y) :|: x1 >= 0, z' = 1 + (1 + x1), z = 1, y >= 0, z'' = 1 + y
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + x2, x), 1 + x2, x), 1 + x) :|: z' = 1 + x, x >= 0, z'' = 1 + (1 + x2), z = 0, x2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + y) :|: z = 1, y >= 0, z'' = 1 + y, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   minus(z, z') -{ 2 }-> if_minus(0, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y


----------------------------------------

(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0


----------------------------------------

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { le }
   { minus, if_minus }
   { gcd, if_gcd }

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}

----------------------------------------

(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}

----------------------------------------

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

----------------------------------------

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

----------------------------------------

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

Computed SIZE bound using CoFloCo for: if_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: ?, size: O(n^1) [z]
  if_minus: runtime: ?, size: O(n^1) [z']

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 4*z + z*z' + 2*z'

Computed RUNTIME bound using KoAT for: if_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 4*z' + z'*z'' + z''

----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {gcd,if_gcd}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

----------------------------------------

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 + (z'' - 2), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {gcd,if_gcd}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: gcd
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

Computed SIZE bound using KoAT for: if_gcd
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z' + z''

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 + (z'' - 2), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {gcd,if_gcd}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  gcd: runtime: ?, size: O(n^1) [z + z']
  if_gcd: runtime: ?, size: O(n^1) [z' + z'']

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: gcd
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 2 + 58*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 58*z' + 26*z'^2 + 4*z'^3

Computed RUNTIME bound using KoAT for: if_gcd
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 8 + 98*z' + 58*z'*z'' + 24*z'*z''^2 + 54*z'^2 + 24*z'^2*z'' + 12*z'^3 + 98*z'' + 54*z''^2 + 12*z''^3

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 + (z'' - 2), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  gcd: runtime: O(n^3) [2 + 58*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 58*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z']
  if_gcd: runtime: O(n^3) [8 + 98*z' + 58*z'*z'' + 24*z'*z''^2 + 54*z'^2 + 24*z'^2*z'' + 12*z'^3 + 98*z'' + 54*z''^2 + 12*z''^3], size: O(n^1) [z' + z'']

----------------------------------------

(33) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(34)
BOUNDS(1, n^3)

----------------------------------------

(35) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   gcd(0, z0) -> z0
   gcd(s(z0), 0) -> s(z0)
   gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
   if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
   if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   GCD(0, z0) -> c7
   GCD(s(z0), 0) -> c8
   GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   GCD(0, z0) -> c7
   GCD(s(z0), 0) -> c8
   GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:   le_2, minus_2, if_minus_3, gcd_2, if_gcd_3

Defined Pair Symbols:   LE_2, MINUS_2, IF_MINUS_3, GCD_2, IF_GCD_3

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5, c6_1, c7, c8, c9_2, c10_2, c11_2


----------------------------------------

(37) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(38)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0, z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   GCD(0, z0) -> c7
   GCD(s(z0), 0) -> c8
   GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

The (relative) TRS S consists of the following rules:

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0, z0) -> 0
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   gcd(0, z0) -> z0
   gcd(s(z0), 0) -> s(z0)
   gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
   if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
   if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(39) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(40)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   LE(0', z0) -> c
   LE(s(z0), 0') -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(0', z0) -> c3
   MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
   IF_MINUS(true, s(z0), z1) -> c5
   IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
   GCD(0', z0) -> c7
   GCD(s(z0), 0') -> c8
   GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

The (relative) TRS S consists of the following rules:

   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(0', z0) -> 0'
   minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
   if_minus(true, s(z0), z1) -> 0'
   if_minus(false, s(z0), z1) -> s(minus(z0, z1))
   gcd(0', z0) -> z0
   gcd(s(z0), 0') -> s(z0)
   gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
   if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
   if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(41) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(42)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
GCD(0', z0) -> c7
GCD(s(z0), 0') -> c8
GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
gcd(0', z0) -> z0
gcd(s(z0), 0') -> s(z0)
gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
GCD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_GCD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c7:c8:c9 -> c3:c4 -> c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s

----------------------------------------

(43) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, MINUS, le, GCD, minus, gcd

They will be analysed ascendingly in the following order:
LE < MINUS
LE < GCD
le < MINUS
MINUS < GCD
le < GCD
le < minus
le < gcd
minus < GCD
minus < gcd

----------------------------------------

(44)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
GCD(0', z0) -> c7
GCD(s(z0), 0') -> c8
GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
gcd(0', z0) -> z0
gcd(s(z0), 0') -> s(z0)
gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
GCD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_GCD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c7:c8:c9 -> c3:c4 -> c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
LE, MINUS, le, GCD, minus, gcd

They will be analysed ascendingly in the following order:
LE < MINUS
LE < GCD
le < MINUS
MINUS < GCD
le < GCD
le < minus
le < gcd
minus < GCD
minus < gcd

----------------------------------------

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12)) -> gen_c:c1:c28_12(n11_12), rt in Omega(1 + n11_12)

Induction Base:
LE(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1)
c

Induction Step:
LE(gen_0':s9_12(+(n11_12, 1)), gen_0':s9_12(+(n11_12, 1))) ->_R^Omega(1)
c2(LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12))) ->_IH
c2(gen_c:c1:c28_12(c12_12))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(46)
Complex Obligation (BEST)

----------------------------------------

(47)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
GCD(0', z0) -> c7
GCD(s(z0), 0') -> c8
GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
gcd(0', z0) -> z0
gcd(s(z0), 0') -> s(z0)
gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
GCD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_GCD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c7:c8:c9 -> c3:c4 -> c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
LE, MINUS, le, GCD, minus, gcd

They will be analysed ascendingly in the following order:
LE < MINUS
LE < GCD
le < MINUS
MINUS < GCD
le < GCD
le < minus
le < gcd
minus < GCD
minus < gcd

----------------------------------------

(48) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(49)
BOUNDS(n^1, INF)

----------------------------------------

(50)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
GCD(0', z0) -> c7
GCD(s(z0), 0') -> c8
GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
gcd(0', z0) -> z0
gcd(s(z0), 0') -> s(z0)
gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
GCD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_GCD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c7:c8:c9 -> c3:c4 -> c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Lemmas:
LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12)) -> gen_c:c1:c28_12(n11_12), rt in Omega(1 + n11_12)


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
le, MINUS, GCD, minus, gcd

They will be analysed ascendingly in the following order:
le < MINUS
MINUS < GCD
le < GCD
le < minus
le < gcd
minus < GCD
minus < gcd

----------------------------------------

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s9_12(n619_12), gen_0':s9_12(n619_12)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s9_12(+(n619_12, 1)), gen_0':s9_12(+(n619_12, 1))) ->_R^Omega(0)
le(gen_0':s9_12(n619_12), gen_0':s9_12(n619_12)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(52)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(0', z0) -> c3
MINUS(s(z0), z1) -> c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(true, s(z0), z1) -> c5
IF_MINUS(false, s(z0), z1) -> c6(MINUS(z0, z1))
GCD(0', z0) -> c7
GCD(s(z0), 0') -> c8
GCD(s(z0), s(z1)) -> c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) -> c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) -> c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(0', z0) -> 0'
minus(s(z0), z1) -> if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) -> 0'
if_minus(false, s(z0), z1) -> s(minus(z0, z1))
gcd(0', z0) -> z0
gcd(s(z0), 0') -> s(z0)
gcd(s(z0), s(z1)) -> if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0))

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
MINUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c:c1:c2 -> c3:c4
IF_MINUS :: true:false -> 0':s -> 0':s -> c5:c6
le :: 0':s -> 0':s -> true:false
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c3:c4 -> c5:c6
GCD :: 0':s -> 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c:c1:c2 -> c7:c8:c9
IF_GCD :: true:false -> 0':s -> 0':s -> c10:c11
c10 :: c7:c8:c9 -> c3:c4 -> c10:c11
minus :: 0':s -> 0':s -> 0':s
c11 :: c7:c8:c9 -> c3:c4 -> c10:c11
if_minus :: true:false -> 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_12 :: c:c1:c2
hole_0':s2_12 :: 0':s
hole_c3:c43_12 :: c3:c4
hole_c5:c64_12 :: c5:c6
hole_true:false5_12 :: true:false
hole_c7:c8:c96_12 :: c7:c8:c9
hole_c10:c117_12 :: c10:c11
gen_c:c1:c28_12 :: Nat -> c:c1:c2
gen_0':s9_12 :: Nat -> 0':s


Lemmas:
LE(gen_0':s9_12(n11_12), gen_0':s9_12(n11_12)) -> gen_c:c1:c28_12(n11_12), rt in Omega(1 + n11_12)
le(gen_0':s9_12(n619_12), gen_0':s9_12(n619_12)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_12(0) <=> c
gen_c:c1:c28_12(+(x, 1)) <=> c2(gen_c:c1:c28_12(x))
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))


The following defined symbols remain to be analysed:
MINUS, GCD, minus, gcd

They will be analysed ascendingly in the following order:
MINUS < GCD
minus < GCD
minus < gcd